Simplify ((2w^3x^-6y^4z)/(8w^-5x^4y^-6z^-3))^-4

${(\frac{2 w^{3} x^{- 6} y^{4} z}{8 w^{- 5} x^{4} y^{- 6} z^{- 3}})}^{- 4}$

Step 1

Move $x^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{2 w^{3} y^{4} z}{8 w^{- 5} x^{4} y^{- 6} z^{- 3} x^{6}})}^{- 4}$

Step 2

Multiply $x^{4}$ by $x^{6}$ by adding the exponents.

Move $x^{6}$ .

${(\frac{2 w^{3} y^{4} z}{8 w^{- 5} (x^{6} x^{4}) y^{- 6} z^{- 3}})}^{- 4}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{2 w^{3} y^{4} z}{8 w^{- 5} x^{6 + 4} y^{- 6} z^{- 3}})}^{- 4}$

Add $6$ and $4$ .

${(\frac{2 w^{3} y^{4} z}{8 w^{- 5} x^{10} y^{- 6} z^{- 3}})}^{- 4}$

Step 3

Move $w^{- 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{2 w^{3} y^{4} z w^{5}}{8 x^{10} y^{- 6} z^{- 3}})}^{- 4}$

Step 4

Move $y^{- 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{2 w^{3} y^{4} z w^{5} y^{6}}{8 x^{10} z^{- 3}})}^{- 4}$

Step 5

Move $z^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{2 w^{3} y^{4} z w^{5} y^{6} z^{3}}{8 x^{10}})}^{- 4}$

Step 6

Simplify the numerator.

Multiply $w^{3}$ by $w^{5}$ by adding the exponents.

Move $w^{5}$ .

${(\frac{2 (w^{5} w^{3}) y^{4} z y^{6} z^{3}}{8 x^{10}})}^{- 4}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{2 w^{5 + 3} y^{4} z y^{6} z^{3}}{8 x^{10}})}^{- 4}$

Add $5$ and $3$ .

${(\frac{2 w^{8} y^{4} z y^{6} z^{3}}{8 x^{10}})}^{- 4}$

Multiply $y^{4}$ by $y^{6}$ by adding the exponents.

Move $y^{6}$ .

${(\frac{2 w^{8} (y^{6} y^{4}) z \cdot z^{3}}{8 x^{10}})}^{- 4}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{2 w^{8} y^{6 + 4} z \cdot z^{3}}{8 x^{10}})}^{- 4}$

Add $6$ and $4$ .

${(\frac{2 w^{8} y^{10} z \cdot z^{3}}{8 x^{10}})}^{- 4}$

Multiply $z$ by $z^{3}$ by adding the exponents.

Move $z^{3}$ .

${(\frac{2 w^{8} y^{10} (z^{3} z)}{8 x^{10}})}^{- 4}$

Multiply $z^{3}$ by $z$ .

Raise $z$ to the power of $1$ .

${(\frac{2 w^{8} y^{10} (z^{3} z^{1})}{8 x^{10}})}^{- 4}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{2 w^{8} y^{10} z^{3 + 1}}{8 x^{10}})}^{- 4}$

Add $3$ and $1$ .

${(\frac{2 w^{8} y^{10} z^{4}}{8 x^{10}})}^{- 4}$

Step 7

Cancel the common factor of $2$ and $8$ .

Factor $2$ out of $2 w^{8} y^{10} z^{4}$ .

${(\frac{2 (w^{8} y^{10} z^{4})}{8 x^{10}})}^{- 4}$

Cancel the common factors.

Factor $2$ out of $8 x^{10}$ .

${(\frac{2 (w^{8} y^{10} z^{4})}{2 (4 x^{10})})}^{- 4}$

Cancel the common factor.

${(\frac{2 (w^{8} y^{10} z^{4})}{2 (4 x^{10})})}^{- 4}$

Rewrite the expression.

${(\frac{w^{8} y^{10} z^{4}}{4 x^{10}})}^{- 4}$

Step 8

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{4 x^{10}}{w^{8} y^{10} z^{4}})}^{4}$

Step 9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{4 x^{10}}{w^{8} y^{10} z^{4}}$ .

$\frac{{(4 x^{10})}^{4}}{{(w^{8} y^{10} z^{4})}^{4}}$

Apply the product rule to $4 x^{10}$ .

$\frac{4^{4} {(x^{10})}^{4}}{{(w^{8} y^{10} z^{4})}^{4}}$

Apply the product rule to $w^{8} y^{10} z^{4}$ .

$\frac{4^{4} {(x^{10})}^{4}}{{(w^{8} y^{10})}^{4} {(z^{4})}^{4}}$

Apply the product rule to $w^{8} y^{10}$ .

$\frac{4^{4} {(x^{10})}^{4}}{{(w^{8})}^{4} {(y^{10})}^{4} {(z^{4})}^{4}}$

Step 10

Simplify the numerator.

Raise $4$ to the power of $4$ .

$\frac{256 {(x^{10})}^{4}}{{(w^{8})}^{4} {(y^{10})}^{4} {(z^{4})}^{4}}$

Multiply the exponents in ${(x^{10})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{256 x^{10 \cdot 4}}{{(w^{8})}^{4} {(y^{10})}^{4} {(z^{4})}^{4}}$

Multiply $10$ by $4$ .

$\frac{256 x^{40}}{{(w^{8})}^{4} {(y^{10})}^{4} {(z^{4})}^{4}}$

Step 11

Simplify the denominator.

Multiply the exponents in ${(w^{8})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{256 x^{40}}{w^{8 \cdot 4} {(y^{10})}^{4} {(z^{4})}^{4}}$

Multiply $8$ by $4$ .

$\frac{256 x^{40}}{w^{32} {(y^{10})}^{4} {(z^{4})}^{4}}$

Multiply the exponents in ${(y^{10})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{256 x^{40}}{w^{32} y^{10 \cdot 4} {(z^{4})}^{4}}$

Multiply $10$ by $4$ .

$\frac{256 x^{40}}{w^{32} y^{40} {(z^{4})}^{4}}$

Multiply the exponents in ${(z^{4})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{256 x^{40}}{w^{32} y^{40} z^{4 \cdot 4}}$

Multiply $4$ by $4$ .

$\frac{256 x^{40}}{w^{32} y^{40} z^{16}}$

Simplify ((2w^3x^-6y^4z)/(8w^-5x^4y^-6z^-3))^-4

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